Flocking of Cucker-Smale model with intrinsic dynamics

Lining Ru; Xiaoping Xue

doi:10.3934/dcdsb.2019168

Article Contents

2019, Volume 24, Issue 12: 6817-6835. Doi: 10.3934/dcdsb.2019168

This issue Previous Article Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink Next Article On global existence and blow-up for damped stochastic nonlinear Schrödinger equation

Flocking of Cucker-Smale model with intrinsic dynamics

Lining Ru^1,2, and
Xiaoping Xue^1, ,

1.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.
Department of Mathematics, Suzhou University of Science and Technology, Suzhou 215009, China

^* Corresponding author: Xiaoping Xue
^* Corresponding author: Xiaoping Xue

Received: August 31, 2018

Revised: January 31, 2019

Early access: July 2019

Published: September 2019

The authors are supported by NSF of China grants 11731010 and 11671109

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Abstract

In this paper, we consider the flocking problem of modified continu- ous-time and discrete-time Cucker-Smale models where every agent has its own intrinsic dynamics with Lipschitz property. The dynamics of the models are governed by the interplay between agents' own intrinsic dynamics and Cucker-Smale coupling dynamics. Based on the explicit construction of Lyapunov functionals, we show that conditional flocking would occur. And then we study the relationship between the Lipschitz constant $ L $ of the intrinsic dynamics and exponent $ \beta $ measuring the strength of the interaction between agents when flocking occurs. We also give two examples to show flocking might not occur for enough large $ L $ or unconditionally for $ \beta>0 $. At last, we provide several numerical simulations to illustrate our theoretical results.

Keywords:

Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 91D30.

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Figure 1. Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 3] $; {(b)} velocity $ V_2 $ for $ t\in [0, \; 3] $

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Figure 2. Evolutions of $ d_V(t) $ and $ d_X(t) $ with 6 particles: {(a)} $ d_V(t) $ for $ t\in [0, \; 3] $; {(b)} $ d_X(t) $ for $ t\in [0, \; 3] $

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Figure 3. Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 150] $; {(b)} velocity $ V_2 $ for $ t\in [0, \; 150] $

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Figure 4. Evolutions of $ d_V[t] $ and $ d_X[t] $ with 6 particles: {(a)} $ d_V[t] $ for $ t\in [0, \; 150] $; {(b)} $ d_X[t] $ for $ t\in [0, \; 150] $

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Figure 5. Dynamics of velocity $ V_2 $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $

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Figure 6. Evolutions of $ d_V[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $

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Figure 7. Evolutions of $ d_X[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $

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